Complete Notes - Magnetic Materials Notes | EduRev

: Complete Notes - Magnetic Materials Notes | EduRev

 Page 1


Chapter 1
Magnetic Materials
1.1 Preliminaries
1.1.1 Required Knowledge
• Magnetism
• Electron spin
• Atom
• Angular momentum (quantum)
• Statistical mechanics
1.1.2 Reading
• Hook and Hall 7.1-7.3, 8.1-8.7
1.2 Introduction
• Magnettechnologyhasmadeenormousadvancesinrecentyears–without
the reductions in size that have come with these advances many modern
devices would be impracticable.
1
Page 2


Chapter 1
Magnetic Materials
1.1 Preliminaries
1.1.1 Required Knowledge
• Magnetism
• Electron spin
• Atom
• Angular momentum (quantum)
• Statistical mechanics
1.1.2 Reading
• Hook and Hall 7.1-7.3, 8.1-8.7
1.2 Introduction
• Magnettechnologyhasmadeenormousadvancesinrecentyears–without
the reductions in size that have come with these advances many modern
devices would be impracticable.
1
2 CHAPTER 1. MAGNETIC MATERIALS
• The important quantity for many purposes is the energy density of the
magnet.
1.3 Magnetic properties - reminder
• There are two ?elds to consider:
– The magnetic ?eld H which is generated by currents according to
Ampère’s law. H is measured in A m
-1
(Oersteds in old units)
– The magnetic induction, or magnetic ?ux density,B, which gives the
energy of a dipole in a ?eld, E =-m.B and the torque experienced
by a dipole momentm asG=m×B. B is measured in Wb m
-2
or
T (Gauss in old units).
Page 3


Chapter 1
Magnetic Materials
1.1 Preliminaries
1.1.1 Required Knowledge
• Magnetism
• Electron spin
• Atom
• Angular momentum (quantum)
• Statistical mechanics
1.1.2 Reading
• Hook and Hall 7.1-7.3, 8.1-8.7
1.2 Introduction
• Magnettechnologyhasmadeenormousadvancesinrecentyears–without
the reductions in size that have come with these advances many modern
devices would be impracticable.
1
2 CHAPTER 1. MAGNETIC MATERIALS
• The important quantity for many purposes is the energy density of the
magnet.
1.3 Magnetic properties - reminder
• There are two ?elds to consider:
– The magnetic ?eld H which is generated by currents according to
Ampère’s law. H is measured in A m
-1
(Oersteds in old units)
– The magnetic induction, or magnetic ?ux density,B, which gives the
energy of a dipole in a ?eld, E =-m.B and the torque experienced
by a dipole momentm asG=m×B. B is measured in Wb m
-2
or
T (Gauss in old units).
1.3. MAGNETIC PROPERTIES - REMINDER 3
• In free space,B = µ
0
H.
• In a material
B = µ
0
(H+M)
= µ
0
µ
r
H
where µ
r
is the relative permeability, ? is the magnetic susceptibility,
which is a dimensionless quantity.
• Note, though, that ? is sometimes tabulated as the molar susceptibility
?
m
= V
m
?,
whereV
m
isthevolumeoccupiedbyonemole, orasthemass susceptibility
?
g
=
?
?
,
where ? is the density.
• M, the magnetisation, is the dipole moment per unit volume.
M= ?H.
• In general, µ
r
(and hence ?) will depend on position and will be tensors
(so thatB is not necessarily parallel toH).
• Even worse, some materials are non-linear, so that µ
r
and ? are ?eld-
dependent.
• The e?ects are highly exaggerated in these diagrams.
Page 4


Chapter 1
Magnetic Materials
1.1 Preliminaries
1.1.1 Required Knowledge
• Magnetism
• Electron spin
• Atom
• Angular momentum (quantum)
• Statistical mechanics
1.1.2 Reading
• Hook and Hall 7.1-7.3, 8.1-8.7
1.2 Introduction
• Magnettechnologyhasmadeenormousadvancesinrecentyears–without
the reductions in size that have come with these advances many modern
devices would be impracticable.
1
2 CHAPTER 1. MAGNETIC MATERIALS
• The important quantity for many purposes is the energy density of the
magnet.
1.3 Magnetic properties - reminder
• There are two ?elds to consider:
– The magnetic ?eld H which is generated by currents according to
Ampère’s law. H is measured in A m
-1
(Oersteds in old units)
– The magnetic induction, or magnetic ?ux density,B, which gives the
energy of a dipole in a ?eld, E =-m.B and the torque experienced
by a dipole momentm asG=m×B. B is measured in Wb m
-2
or
T (Gauss in old units).
1.3. MAGNETIC PROPERTIES - REMINDER 3
• In free space,B = µ
0
H.
• In a material
B = µ
0
(H+M)
= µ
0
µ
r
H
where µ
r
is the relative permeability, ? is the magnetic susceptibility,
which is a dimensionless quantity.
• Note, though, that ? is sometimes tabulated as the molar susceptibility
?
m
= V
m
?,
whereV
m
isthevolumeoccupiedbyonemole, orasthemass susceptibility
?
g
=
?
?
,
where ? is the density.
• M, the magnetisation, is the dipole moment per unit volume.
M= ?H.
• In general, µ
r
(and hence ?) will depend on position and will be tensors
(so thatB is not necessarily parallel toH).
• Even worse, some materials are non-linear, so that µ
r
and ? are ?eld-
dependent.
• The e?ects are highly exaggerated in these diagrams.
4 CHAPTER 1. MAGNETIC MATERIALS
1.4 Measuring magnetic properties
1.4.1 Force method
• Uses energy of induced dipole
E =-
1
2
mB =-
1
2
µ
0
?VH
2
,
so in an inhomogeneous ?eld
F =-
dE
dx
=
1
2
µ
0
V?
dH
2
dx
= µ
0
V?H
dH
dx
.
• Practically:
– set up large uniformH;
– superpose linear gradient with additional coils
– vary second ?eld sinusoidally and use lock-in ampli?er to measure
varying force
1.4.2 Vibrating Sample magnetometer
• oscillate sample up and down
• measure emf induced in coils A and B
Page 5


Chapter 1
Magnetic Materials
1.1 Preliminaries
1.1.1 Required Knowledge
• Magnetism
• Electron spin
• Atom
• Angular momentum (quantum)
• Statistical mechanics
1.1.2 Reading
• Hook and Hall 7.1-7.3, 8.1-8.7
1.2 Introduction
• Magnettechnologyhasmadeenormousadvancesinrecentyears–without
the reductions in size that have come with these advances many modern
devices would be impracticable.
1
2 CHAPTER 1. MAGNETIC MATERIALS
• The important quantity for many purposes is the energy density of the
magnet.
1.3 Magnetic properties - reminder
• There are two ?elds to consider:
– The magnetic ?eld H which is generated by currents according to
Ampère’s law. H is measured in A m
-1
(Oersteds in old units)
– The magnetic induction, or magnetic ?ux density,B, which gives the
energy of a dipole in a ?eld, E =-m.B and the torque experienced
by a dipole momentm asG=m×B. B is measured in Wb m
-2
or
T (Gauss in old units).
1.3. MAGNETIC PROPERTIES - REMINDER 3
• In free space,B = µ
0
H.
• In a material
B = µ
0
(H+M)
= µ
0
µ
r
H
where µ
r
is the relative permeability, ? is the magnetic susceptibility,
which is a dimensionless quantity.
• Note, though, that ? is sometimes tabulated as the molar susceptibility
?
m
= V
m
?,
whereV
m
isthevolumeoccupiedbyonemole, orasthemass susceptibility
?
g
=
?
?
,
where ? is the density.
• M, the magnetisation, is the dipole moment per unit volume.
M= ?H.
• In general, µ
r
(and hence ?) will depend on position and will be tensors
(so thatB is not necessarily parallel toH).
• Even worse, some materials are non-linear, so that µ
r
and ? are ?eld-
dependent.
• The e?ects are highly exaggerated in these diagrams.
4 CHAPTER 1. MAGNETIC MATERIALS
1.4 Measuring magnetic properties
1.4.1 Force method
• Uses energy of induced dipole
E =-
1
2
mB =-
1
2
µ
0
?VH
2
,
so in an inhomogeneous ?eld
F =-
dE
dx
=
1
2
µ
0
V?
dH
2
dx
= µ
0
V?H
dH
dx
.
• Practically:
– set up large uniformH;
– superpose linear gradient with additional coils
– vary second ?eld sinusoidally and use lock-in ampli?er to measure
varying force
1.4.2 Vibrating Sample magnetometer
• oscillate sample up and down
• measure emf induced in coils A and B
1.5. EXPERIMENTAL DATA 5
• compare with emf in C and D from known magnetic moment
• hence measured sample magnetic moment
1.5 Experimental data
• In the ?rst 60 elements in the periodic table, the majority have negative
susceptibility – they are diamagnetic.
1.6 Diamagnetism
• Classically, we have Lenz’s law, which states that the action of a magnetic
?eld on the orbital motion of an electron causes a back-emf which opposes
the magnetic ?eld which causes it.
• Frankly, this is an unsatisfactory explanation, but we cannot do better
until we have studied the inclusion of magnetic ?elds into quantum me-
chanics using magnetic vector potentials.
• Imagine an electron in an atom as a charge e moving clockwise in the x-y
plane in a circle of radius a, area A, with angular velocity ?.
• This is equivalent to a current
I =charge/time= e?/(2p),
so there is a magnetic moment
µ= IA= e?a
2
/2.
• The electron is kept in this orbit by a central force
F = m
e
?
2
a.
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